Assuming your interest is in constant mean curvature surfaces with circular boundary, I found the survey, "Surfaces with constant mean curvature in Euclidean space" by R. Lopez to be a great introduction, and it contains the state of the art, and several references.
Hopf proved that the only constant mean curvature closed surfaces of genus 0 are spheres. Alexandrov's theorem says that constant mean curvature closed surfaces that are embedded are spheres. Unfortunately when we allow boundaries both analogs are conjectural:
Conjecture 1: The only constant mean curvature compact surfaces with circular boundary that are topological disks are spherical caps.
Conjecture 2: The only constant mean curvature compact surfaces with circular boundary that are embedded are spherical caps.
